The Micro and Macro of (Unconventional) Monetary Policy

Gregor Boehl
Uni Bonn
Link to slides: https://gregorboehl.com/live/xrs_revealjs

Motivation

  • Deposit & lending rates not directly controled by ECB
  • But central in macroeconomic theory: $$\small c_t = E_t c_{t+1} - \frac{1}{\sigma} (\color{lightgreen}r_t \color{white}- E_t \pi_{t+1})$$
    Wanted:
  • A unifying theory of the role of reserves, bank deposits & interest rates in general equilibrium
  • Implications on monetary & fiscal policy

This Paper

  • Develop an IO-model of banking sector with liquidity friction
  • Banks demand reserves to hedge liquidity risk
  • Embedded into a medium-scale NK framework

Policy Effectiveness

IOR policy reserves policy
MRR* binding low high
MRR* slack high low
sign of effect always positive state dependend:
depends on investment elasticity
*Minimal reserve requirement

Model:

An IO model of the banking sector

Banking sector

  • banks hold assets and reserves, backed by deposits: $$ A_{i,t} + J_{i,t} = D_{i,t} $$
  • assets are fixed at the beginning of each period
  • deposits are subject to random transfers (prob. $\chi$) $$ \Delta {D}_{i,t} \sim \mathcal{N} \left( 0, \frac{{D}_{i,t} D_{-i,t}}{\sum D_{j,t}} (2 \chi - \chi^2)\right). $$
  • any transfered unit of deposits not backed by reserves incurs a cost $\gamma$

Banking sector: bank $i$'s problem

  • Expected cost for each bank: $$ g(J_{i,t}, D_{i,t}) = h(D_{i,t}) f_h\left(J_{i,t}\right) - J_{i,t} \left[1-F_h\left(J_{i,t}\right)\right], $$
  • Banks are subject to a minimal reserve requirement (MRR) $$ J_{i,t} \geq \psi D_{i,t} $$
  • Each bank $i$ maximizes profits s.t. balance sheet, expected costs, MRR & competitors actions

Banking sector: aggregation

  • IOR rate $R^j$, deposit rate $R$, lending rate $R^b$
  • Reserves & deposits scaled by liquidity risk $\hat{J} = \frac{J}{\nu}$
  • $\hat{f} = f\left(\hat{J}| 0, \hat{D}\right)$, $\hat{f}_\psi = f\left(\psi\hat{D}| 0, \hat{D}\right)$, $\hat{F}=...$


FOCs:
$$ \small \begin{align} R = (1-\psi) R^b + \psi R^j &-\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \color{gray} \underbrace{ \color{white} R^j - R^b + \gamma \left[1- \hat{F} \right] }_{MPJ} &= \min\left\{ 0, \; R^j - R^b + \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$
FOCs:
$$ \small \begin{align} R = (1-\psi) R^b + \psi R^j &-\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \underbrace{ R^j - R^b + \gamma \left[1- \hat{F} \right] }_{MPJ} &= \min\left\{ 0, \; R^j - R^b + \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$
FOCs when MRR slack:
$$ \small \begin{align} R = \color{gray}(1-\psi)\color{white} R^b \color{gray}+ \psi R^j \color{white} &-\gamma \color{gray} \left( \color{white} 0.5\hat{f} \color{gray} - \psi \left[1-\hat{F}\right] \right) \\ \color{white} \underbrace{ R^j - R^b + \gamma \left[1-\hat{F}\right] }_{MPJ} &= \color{gray} \min\left\{ \color{white} 0 \color{gray}, \; R^j - R^b + \gamma \left[1-\hat{F}_\psi \right] \right\} \end{align} $$
FOCs at MRR:
$$ \small \begin{align} R = (1-\psi) R^b + \psi R^j &-\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \color{gray} \underbrace{ R^j - R^b + \gamma \left[1- \color{white} \hat{F} \color{gray} \right]}_{MPJ} &= \color{gray} \min\left\{ 0, \; R^j - R^b + \gamma \left[1- \color{white} \hat{F}_\psi \color{gray} \right] \right\} \end{align} $$

Banking sector: partial equilibrium

$$ \scriptsize \begin{align} {d_A}\left(R^b\right) &= \hat{D}, \; \left\{\hat{J}, R^j\right\} \; \text{fixed}\\ \gamma \left[1- \hat{F} \right] &= \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$
$$ \scriptsize \begin{align} {d_A}\left(R^b\right) &= \hat{D}, \; \left\{\hat{J}, R^j\right\} \; \text{fixed}\\ \color{gray} \gamma \left[1- \color{white} \hat{J} \color{gray} \right] &= \color{gray} \min\left\{ R^b - R^j, \; \gamma \left[1- \color{white} \psi \hat{D} \color{gray} \right] \right\} \end{align} $$
$$ \scriptsize \begin{align} {d_A}\left(R^b\right) &= \hat{D}, \; \left\{\hat{J}, R^j\right\} \; \text{fixed}\\ \color{white} \gamma \left[1-\hat{F}\right] &= \color{gray}\min\left\{ \color{white} R^b - R^j \color{gray}, \; \gamma \left[1-\hat{F}_\psi \right] \right\} \end{align} $$
$$ \scriptsize \begin{align} {d_A}\left(R^b\right) &= \hat{D}, \; \left\{\hat{J}, R^j\right\} \; \text{fixed}\\ \gamma \left[1- \hat{F} \right] &= \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$

Results:

partial equilibrium

partial equilibrium: reserves policy


$$ \scriptsize \begin{align} R &= (1-\psi) R^b + \psi R^j -\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \gamma \left[1- \hat{F} \right] &= \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$

partial equilibrium: higher elasticity

$$ \scriptsize \begin{align} R &= (1-\psi) R^b + \psi R^j -\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \gamma \left[1- \hat{F} \right] &= \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$

partial equilibrium: IOR policy

$$ \scriptsize \begin{align} R &= (1-\psi) R^b + \psi R^j -\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \gamma \left[1- \hat{F} \right] &= \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$

Results:

general equilibrium

general equilibrium: model

  • Embed banking sector into medium scale DSGE
  • Features:
    capital, capital adjustment, capital utilization, sticky prices รก la Rotemberg, debt financed transfers
  • Fact: determined under IOR & reserves peg (no Taylor principle)
  • Nonlinear perfect foresight solution (using econpizza)

general equilibrium

reserves policy & the MRR

general equilibrium

reserves policy if MRR is slack

general equilibrium

reserves policy @ the MRR

general equilibrium

IOR policy: @ vs. slack MRR

Conclusion

  • Provide a unifying theory of the relationship of
    • bank deposits
    • central bank reserves
    • minimal reserves requirements
    • deposits, lending & IOR rate
  • Effect of reserves policy is highly state dependent